Question

In a right-angled triangle ABC, ∠B = 90°, BD = 3, DC = 4, and AC = 13. A point D is inside the triangle such as ∠BDC = 90°.

Find the values of 2 tan ∠BAC - sin ∠BCD

Answer 1

ΔBDC is a right-angled triangle.
∴ BC²
= BD² +DC²
= 3² + 4²
= 9 + 16
= 25
⇒ BC = 5cm
ΔABC is a right-angled triangle.
∴ AB²
= AC² - BC²
= 13² - 5²
= 169 - 25
= 144
⇒  AB = 12cm
2 tan ∠BAC - sin ∠BCD

= `2 xx "BC"/"AB" - "BD"/"BC"`

= `2 xx (5)/(12) - (3)/(5)`

= `(5)/(6) - (3)/(5)`

= `(25 - 18)/(30)`

= `(7)/(30)`.